Phase-field approach to polycrystalline solidification
including heterogeneous and homogeneous nucleation
Tamás Pusztai,† György Tegze,‡ Gyula I. Tóth,† László Környei,† Gurvinder Bansel,‡ Zhungyun Fan,‡
and László Gránásy‡§
† Research Institute for Solid State Physics and Optics, P O Box 49, H−1525 Budapest, Hungary
‡ Brunel Centre for Advanced Solidification Technology, Brunel University, Uxbridge, UB8 3PH, UK
§ Corresponding author: Laszlo.Granasy@brunel.ac.uk or grana@szfki.hu
Abstract. Advanced phase-field techniques have been applied to address various aspects of polycrystalline solidification including different modes of crystal nucleation. The height of the nucleation barrier has been determined by solving the appropriate Euler-Lagrange equations. The examples shown include the comparison of various models of homogeneous crystal nucleation with atomistic simulations for the single component hard-sphere fluid. Extending previous work for pure systems (Gránásy L, Pusztai T, Saylor D and Warren J A 2007 Phys. Rev. Lett. 98 art no 035703),
heterogeneous nucleation in unary and binary systems is described via introducing boundary conditions that realize the
desired contact angle. A quaternion representation of crystallographic orientation of the individual particles (outlined in
Pusztai T, Bortel G and Gránásy L 2005 Europhys. Lett. 71 131) has been applied for modeling a broad variety of polycrystalline structures including crystal sheaves, spherulites and those built of crystals with dendritic, cubic, rhombododecahedral, truncated octahedral growth morphologies. Finally, we present illustrative results for dendritic polycrystalline solidification obtained using an atomistic phase-field model.
PACS numbers: 61.72.Bb, 61.72.Mm, 64.60.Qb, 64.70.Dv, 81.10.Aj
1. Introduction
A substantial fraction of the technical materials used in the everyday life are polycrystalline, i.e., are composed of crystallites whose size, shape and composition distributions determine their macroscopic properties and failure characteristics of these substances (Cahn 2001). Size of the constituent crystallites may range from nanometers to centimeters in
different classes of materials. While polycrystalline materials are the subject of an intensive research for some time,
many aspects of polycrystalline solidification are still little understood. The complexity of multigrain crystallization is
exemplified by thin polymer layers, which show an enormous richness of crystallization morphologies (Geil 1996). Polycrystalline morphologies of particular interest are the ubiquitous multi-grain dendritic and spherulitic structures. The
multi-grain dendritic structures are composed of a large number of pine-tree-like dendritic crystals (relatives of the iceflowers forming on window panes) and besides a broad range of other materials they have been seen in crystallizing
colloidal suspensions (Cheng et al 2002). The term ‘spherulite’ is used in a broader sense for densely branched, polycrystalline solidification patterns (Magill 2001). Besides polymers and biopolymers, they have been seen in a broad
variety of systems including alloys, mineral aggregates and volcanic rocks, liquid crystals, oxide and metallic glasses,
even chocolate and biological systems. In particular, the world of minerals provides beautifully complex examples of
such structures (Shelf and Hill 2003). Appearance of semi-crystalline spherulites of amyloid fibrils is associated with
the Alzheimer and Creutzfeldt-Jakob diseases, type II. diabetes, and a range of systemic and neurotic disorders (Jin et al
2003, Krebs et al 2005), kidney stones of polycrystalline spherulitic structure have been observed (Khan et al 1979,
Lambert et al 1998), and the formation kinetics of ice crystals influence the extent of damage biological tissues undergo
during freezing (Zacharaissen and Hammel 1988). Other remarkably complex polycrystalline morphologies appear in
composite materials, such as the "shish-kebab" structure in carbon nanotube containing polymers (Li et al 2006), and
the plate like branched structures (e.g, graphite in cast iron and in other systems (Napolitano et al 2004, Hyde et al
2004)). Crystallization can be influenced by intrinsic and external fields such as composition, temperature, pressure,
flow and electromagnetic fields. For example, modulated fields have been used to influence the dendritic crystallization
morphology both in experiment and modeling: flow (Bouissou et al 1990), laser pulses (Quian and Cummins 1990,
Murray et al 1995), and pressure (Börzsönyi et al 1999, 2000, Koss et al 2005, Li et al 2007). Although in the present
paper, we concentrate mainly on techniques that are able to address polycrystalline solidification in specific intrinsic
(composition) and external fields (temperature), we consider possible inclusion of other fields (e.g., flow).
First, we need, however, a suitable model of polycrystalline solidification that incorporates crystal nucleation and
growth on equal footing. The fact that very similar polycrystalline morphologies are seen in substances of very different
molecular geometry raises the hope that a coarse grained field theoretic model that neglects the molecular details might
be able to capture some of the essential factors that govern crystalline pattern formation in such systems. It is expected
that nucleation, diffusional instabilities, crystal symmetries, and the presence of particulate impurities play an important
role. A particularly interesting mode of polycrystalline solidification, identified recently, is growth front nucleation
1
(GFN), where growth takes place via continuous formation of new grains at the solidification front, a growth mechanism typical for spherulites and fractal-like polycrystals (Gránásy et al 2004a 2004b 2005). Accordingly, the model
needs to address homogeneous and heterogeneous nucleation of growth centers and growth front nucleation (homogeneous and heterogeneous) together with diffusional instabilities.
Advances in computational materials science offer various methods to model polycrystalline solidification, which
include cellular automata (e.g., Zhu and Hong 2002, Beltram-Sanchez and Stefanescu 2004, Zhu et al 2008), level set
(e.g., Tryggvason et al 2001, Tan and Zabaras 2006 2007) and other front tracking techniques (e.g., Schmidt 1996,
Steinbach et al 1999, Jacot and Rappaz 2002), and phase-field approaches (recent reviews: Boettinger et al 2002, Chen
2002, Hoyt et al 2003, Gránásy et al 2004a). Among them the phase-field models appear to be perhaps the most popular
ones as they connect thermodynamic and kinetic properties with microstructure via a transparent mathematical formalism. In the phase-field theory, the local state of matter is characterized by a non-conserved structural order parameter
φ(r, t), called phase field, which monitors the transition between the solid and liquid states. The time evolution of the
structural order parameter is usually coupled to that of other slowly evolving conserved fields such as temperature or
composition.
The phase-field model has already been used to determine the height of the nucleation barrier for homogeneous and
heterogeneous nucleation (Gránásy et al 2002 2003a 2007) In the case of homogeneous nucleation a quantitative study
has been performed for the hard sphere systems utilizing the thermodynamic, interfacial free energy and interface thickness data to fix the model parameters in equilibrium. Then the Euler-Lagrange equations have been solved to obtain the
unstable equilibrium corresponding to crystal nuclei in the supersaturated state. This procedure delivers the free energy
of nuclei without adjustable parameters, which can be then compared to the nucleation barrier data measured directly
by atomistic simulations (Auer and Frenkel 2001a 2001b). It has been found that with an orientation averaged interfacial free energy of ~ 0.61 kT/σ 2, obtained from molecular dynamics and Monte Carlo techniques (Davidchack and
Laird 2000, Cacciuto et al 2003), a fair agreement can be seen with a phase-field model that relies on a quartic free
energy and the usual intuitive but thermodynamically consistent interpolation function. Remarkably, recently, the free
energy of the hard sphere crystal-liquid interface has been reduced considerably to ~ 0.559 kT/σ 2 (Davidchack et al
2006) which certainly spoils the fair agreement (here k, T, and σ are Boltzmann's constant, the temperature and the diameter of the hard spheres, respectively). It would be then natural to explore whether a better approximation could be
obtained by using physically motivated double well and interpolation functions emerging from a Ginzburg-Landau expansion of the free energy (Gránásy and Pusztai 2002).
In the case of heterogeneous nucleation appropriate boundary conditions have been introduced at the foreign wall to
realize the required contact angle (Gránásy et al 2007). Properties of the heterogeneous nuclei in two dimensions (2D)
were obtained by solving numerically the respective Euler-Lagrange equation under these boundary conditions. This
work needs too be extended to 3D and to alloys.
Modeling of polycrystalline solidification requires the inclusion of homogeneous and/or heterogeneous nucleation
into the phase-field model. In field theoretic models it is done traditionally by adding Langevin noise of appropriate
properties to the equations of motion (Gunton et al 1983). However, to describe the impingement of a large number of
crystallites that grow anisotropically, one needs to incorporate the crystallographic orientations that allow the specification of the preferred growth directions of growth. The first phase-field model that introduces different crystallographic
orientations into a solidifying system (Morin et al 1995) relies on a free energy density that has n wells, corresponding
to n crystallographic orientations, breaking thus the rotational symmetry of the free energy. Simulations have been then
performed to study polymorphous crystallization, where the composition of the liquid remains close to that of the crystal, therefore, chemical diffusion plays a minor role, and the system follows the Johnson-Mehl-Avrami-Kolmogorov
(JMAK) kinetics (see e.g., Christian 1981). A weakness of the model is that the rotational invariance of the free energy
density had to be sacrificed and a finite number of crystallographic orientations need to be introduced to enable the formation of grain boundaries of finite thickness.
A different approach for addressing the formation of particles with random crystallographic orientations is realized
by the multi-phase-field theory (MPFT; see e.g., Steinbach et al 1996, Fan and Chen 1996, Tiaden et al 1998, Diepers et
al 2002, Krill and Chen 2002), in which a separate phase field is introduced for every crystal grain. This models offer
flexibility at the expense of enhanced mathematical/numerical complexity. MPFT has been used to study polycrystalline
dendritic and eutectic/peritectic solidification, and has also been successfully applied for describing the time evolution
of multigrain structures. However, the large number of phase fields applied in these approaches leads to difficulties,
when nucleation is to be modeled by Langevin noise. While noise-induced nucleation can certainly be substituted by
inserting nuclei by ‘hand’ into the simulations, this procedure becomes excessively non-trivial, when structures that
require the nucleation of different crystallographic orientations at the growth front are to be addressed. Such a treatment, furthermore, rules out any possible interaction between diffusion and the orientation of new grains. In this way
realization of growth front nucleation in the MPFT is not immediately straightforward.
It appears that modeling of complex polycrystalline structures and especially of GFN, requires another approach that
relies on an orientation field to monitor the crystallographic orientation. The first model of this kind has been put forward by Kobayashi et al (1998) to model polycrystalline solidification in 2D, which uses a non-conserved scalar field to
monitor crystallographic orientation. Assuming a free energy density of fori = HT∇θ, where the coefficient H has a
2
minimum at the position of the interface, the minimization of free energy leads to a stepwise variation of θ(r), a behavior approximating reasonably the experimental reality of stable, flat grain boundaries. (Such minimum can be realized
making the coefficient H dependent on the phase field, e.g., by introducing the factor 1 − p(φ) into fori (Gránásy et al
2002)) Various modifications of this approach have been successfully applied for describing problems including solidsolid and solid-liquid interfaces (Kobayashi et al 1998, 2000, Warren et al 2003). A further important contribution was
the modeling of the nucleation of grains with different crystallographic orientations, which has been solved by Gránásy
et al (2002), who extended the orientation field θ into the liquid phase, where it has been made to fluctuate in time and
space. Assigning local crystal orientation to liquid regions, even a fluctuating one, may seem artificial at first sight.
However, due to geometrical and/or chemical constraints, a short-range order exists even in simple liquids, which is
often similar to the one in the solid. Rotating the crystalline first-neighbor shell so that it aligns optimally with the local
liquid structure, one may assign a local orientation to every atom in the liquid. The orientation obtained in this manner
indeed fluctuates in time and space. The correlation of the atomic positions/angles shows how good this fit is. (In the
model, the fluctuating orientation field and the phase field play these roles.) Approaching the solid from the liquid, the
orientation becomes more definite (the amplitude of the orientational fluctuations decreases) and matches to that of the
solid, while the correlation between the local liquid structure and the crystal structure improves. fori = [1 − p(φ)]∇θ
recovers this behavior by realizing a strong coupling between the orientation and phase fields. This addition to the
orientation field model, first introduced by Gránásy et al (2002), facilitates the quenching of orientational defects into
the crystal, leading to a mechanism generating new grains at the growth front. Indeed this approach of ours successfully
describes the formation of such complex polycrystalline growth patterns formed by GFN as disordered (“dizzy”) dendrites (Gránásy et al 2003b), spherulites (Gránásy et al 2003c 2004a 2004b 2005), ‘quadrites’ (Gránásy et al 2005),
fractallike aggregates (2004a) and eutectic grains with preferred orientation between the two crystalline phases (Lewis
et al 2004). The generalization of this approach to three dimensions has been done somewhat later. Practically at the
same time two essentially equivalent formulations have been put forward: Pusztai et al (2005a 2005b) used the quaternion representation for the crystallographic orientation in solidification problems, while Kobayashi and Warren (2005a
2005b) proposed a rotation matrix representation to address grain boundary dynamics. A shortcoming of these earlier
works is that crystal symmetries have not been taken into account in the simulations, although Pusztai et al (2005a
2005b) outlined in their papers how crystal symmetries should be handled in grain boundary formation.
A promising new field theoretic formulation of polycrystalline solidification is the Phase-Field Crystal (PFC) model
(Elder et al 2002 2007, Elder and Grant 2004), which addresses freezing on the atomistic/molecular scale. The PFC
approach is a close relative of the classical density functional theory (DFT) of crystallization: one may derive it by making a specific approximation for the two-particle direct correlation function of the liquid (Elder and Grant 2004, Elder et
al 2007) in the Ramakrishnan-Yussouff expansion of the free energy functional of the crystal relative to the homogeneous liquid (for review on DFT see Oxtoby 1991). Remarkably, the PFC description includes automatically the elastic
effects and crystal anisotropies, while addressing interfaces, dislocations and other lattice defects on the atomic scale. It
has the advantage over traditional atomistic simulations (such as molecular dynamics), that it works on the diffusive
time scale, i.e., processes taking place on about a million times longer time scale than molecular dynamics can address.
The PFC method has already demonstrated its high potential for modeling dendrites, eutectic structures, polycrystalline
solidification, grain boundaries / dislocations, epitaxial growth, crack formation, etc. (Elder and Grant 2004, Elder et al
2007, Provatas et al 2007). However, due to its atomistic nature it cannot be easily used to model large scale polycrystalline structures. Combination of a coarse grained formulation of the binary PFC theory based on the renormalization
group technique outlined for the single component case with adaptive mesh techniques (Goldenfeld et al 2005, Athreya
et al 2006, 2007) will certainly enhance the simulation domain for multi-component systems in the future. Another difficulty is that the crystal lattice and the respective anisotropy of the interfacial free energy cannot be easily tuned, although recent work incorporating three-body correlation opens up the way for advance in this direction (Tupper and
Grant 2008). While the PFC is undoubtedly an excellent tool for investigating the atomistic aspects of polycrystalline
solidification, it cannot easily address such scale morphologies as 3D multi-grain dendritic structures or spherulites:
they seem to belong yet to the domain of conventional phase-field modeling. With appropriate numerical techniques,
however, the PFC model might be applicable to address even such problems under specific conditions in 2D.
Herein, we apply the phase-field method to address various aspects of nucleation and polycrystalline solidification:
(i) We reassess phase-field models of homogeneous crystal nucleation in the hard sphere system. (ii) We determine the
structure and the barrier height for heterogeneous nucleation in a binary alloy. (iii) We apply the model of Pusztai et al
(2005a 2005b) for describing polycrystalline solidification while considering crystal symmetries in handling the orientation field (crystallites with orientations related to each other by symmetry operations should not form grain boundaries),
and demonstrate that the model is able to describe complex polycrystalline solidification morphologies based on dendritic, cubic, rhombododecahedral, and truncated octahedral growth forms, besides the transition between single needle
crystals and polycrystalline spherulites. We combine the model with boundary conditions that realize pre-defined contact angles which is then used to model the formation of shish-kebab structures on nano-fibers. We introduce then a
spatially homogeneous flow and a fixed temperature gradient to mimic directional solidification, which is then used to
model the columnar to equiaxed transition in a binary alloy. (iv) Finally, we model multi-grain dendritic solidification
in the framework of the binary PFC approach.
3
2. Phase-field models used
2.1 Phase-field approach to nucleation barrier in homogeneous and heterogeneous nucleation
As in other continuum models the critical fluctuation or nucleus represents an extremum of the appropriate free energy
functional, therefore can be found by solving the respective sets of Euler-Lagrange equations. In the following we
present phase-field models for two cases: (a) Homogeneous nucleation in the hard sphere system that crystallizes to the
fcc (face centered cubic) structure, where besides the structural changes, we explicitly incorporate the density change
during crystallization. (b) Heterogeneous nucleation in a binary system, where appropriate boundary conditions will be
introduced to fix the contact angle in equilibrium.
2.1.1 Phase-field model of homogeneous nucleation in the hard sphere system Here we consider two possible phasefield approaches. Following previous work (Gránásy et al 2003a), the grand potential of the inhomogeneous system
relative to the initial liquid is assumed to be a local functional of the phase field m monitoring the liquid-solid transition
(m = 0 and 1 in the liquid and in the solid, respectively) and the volume fraction ϕ = (π/6)σ3ρ (here ρ is the number
density of the hard spheres):
 ε 2T
∆Ω = ∫ d 3 r 
 2

(∇m )2 + ∆ω (m,ϕ ) ,

(1)
where, ε is a coefficient that can be related to the interfacial free energy and the interface thickness, T is the temperature, while ∆ ω(m,...) is the local grand free energy density relative to the initial state (that includes the Lagrange multiplier term, ensuring mass conservation; here the Lagrange multiplier is related to the chemical potential of the initial liquid). The gradient term leads to a diffuse crystal-liquid interface, a feature observed both in experiment (e.g, Howe
1996, Huisman et al 1997, Howe and Saka 2004, van der Veen and Reichert 2004) and computer simulations (e.g,
Broughton and Gilmer 1986, Laird and Haymet 1992, Davidchack and Laird 1998, Ramalingam et al 2002). In the
present work, grand potential density is assumed to have the following simple form:
∆ω(m,ϕ) = wTg(m) + [1 − p(m)]fS(ϕ) + p(m)fL(ϕ) − {∂fL/∂ϕ}(ϕ∞)[ϕ − ϕ∞] − fL(ϕ∞),
(2)
where fS(ϕ) and fL(ϕ) are the Helmholtz free energy densities for the solid and liquid states, while ϕ∞ is the volume fractions of the initial (supersaturated) liquid phase. Different “double well” g(m) and “interpolation” functions p(m) will be
used as specified below. The free energy scale w determines the height of the free energy barrier between the bulk solid
and liquid states. Once the functional forms of g(m) and p(m) are specified, model parameters ε and w can be expressed
in terms of γ∞ and the thickness δ of the equilibrium planar interface (Cahn and Hilliard 1958).
Here we use two sets of these functions. One of them has been proposed intuitively in an early formulation of the
PFT and in use widely:
(a) The "standard" set (PFT/S): These functions are assumed to have the form g(φ) = 1/4 φ2(1 − φ)2 and p(φ) = φ3(10
− 15φ + 6φ2), respectively, that emerge from an intuitive formulation of the PFT (Wang et al 1993). Here φ = 1 − m is
the complementing phase field, defined so that it is 0 in the solid and 1 in the liquid. The respective expressions for the
model parameters are as follows: εS2 = 6⋅21/2γ∞δ/Tf, and wS = 6⋅21/2γ∞/(δ⋅Tf). This model has been discussed in detail in
(Gránásy et al 2003a).
(b) Ginzburg-Landau form for fcc structure (PFT/GL): Recently, we have derived these functions for bcc (base cetered cubic) and fcc (face centered cubic) structures (Gránásy and Pusztai 2002) on the basis of a single-order-parameter
Ginzburg-Landau (GL) expansion that considers the crystal symmetries (Shih et al 1987). This treatment yields g(m) =
(1/6)(m2 − 2m4 + m6) and p(m) = 3m4 − 2m6 for the fcc structure, while the expressions that relate the model parameters
to measurable quantities are as follows: εGL2 = (8/3)CεS2, w,GL = wS(4C)−1, where C = ln(0.9/0.1) [3ln(0.9/0.1) −
ln(1.9/1.1)]−1. Combination of the latter double well and interpolation functions with equation (2) is a new construction,
presented here for the first time. Therefore, though it is analogous to the procedure applied in a previous work (Gránásy
et al 2003a), we briefly outline the way the properties of nuclei are determined in this case:
The field distributions, that extremize the free energy, can be obtained solving the appropriate Euler-Lagrange (EL)
equations:
∂I
δΩ ∂I
=
−∇
=0,
δm ∂m
∂∇m
and
∂I
δΩ ∂I
=
−∇
=0,
δϕ ∂ϕ
∂∇ϕ
(3a,b)
where δΩ/δm and δΩ/δϕ stands for the first functional derivative of the grand free energy with respect to the fields m
and ϕ, respectively. Here, I = ½ε 2T(∇m)2 + f(m, ϕ) + λϕ is the total free energy density inclusive the term with a Lagrange multiplier λ ensuring mass conservation, while the Helmholtz free energy density is f(m, ϕ) = wTg(m) + [1 −
4
p(m)] fS(ϕ) + p(m) fL(ϕ). For the sake of simplicity, we assume here an isotropic interfacial free energy (a reasonable
approximation for simple liquids). Note that due to a lack of a gradient term for the field ϕ in the grand potential, equation (3b) yields an implicit relationship between m and ϕ, which can be then inserted into equation (3a), when solving it.
Herein, equation (3b) has been solved numerically, using a variable fourth/fifth order Runge-Kutta method (Korn
and Korn 1970), assuming an unperturbed liquid (m = 0, ϕ = ϕ∞) in the far field (r → ∞), while, for symmetry reasons, a
zero field gradient applies at the center of the fluctuations. Since m and dm/dr are fixed at different locations, the central
value of m that leads to m → m∞ = 0 for r → ∞, have been determined iteratively. Having determined the solutions m(r)
and ϕ(r), the work of formation of the nucleus, W has been obtained by inserting these solutions into the grand potential
difference (equation (1)).
Of these two phase-field models (PFT/S and PFT/GL), the latter, which relies on the Ginzburg-Landau expansion,
incorporates a more detailed physical information on the system (e.g., crystal structure), therefore, it is expected to provide a better approximation to the atomistic simulations.
The physical properties, we use here, are the same as in a previous work of us (Gránásy et al 2003a), with the exception of the 10%-90% interface thickness, which is now allowed to change between 3.0σ and 3.3σ, values that are consistent with the interfacial profiles for a variety of physical properties (such as coarse grained density, diffusion, and
orientational order parameters q4 and q6) at the equilibrium solid-liquid interface of the hard sphere system (Davidchack
and Laird 1998). In Section 3.1, we are going to address uncertainties associated with the interface thickness and interfacial free energy taken from atomistic simulations.
2.1.2 Phase-field model of heterogeneous nucleation in binary alloys Here, we have two fields to describe the local
state of the matter, the usual phase field φ(r) and the concentration field c(r). In the order to keep the problem mathematically simple, we assume again an isotropic solid-liquid interface. Then the Euler-Lagrange equation can be solved
in a cylindrical coordinate system. Furthermore, if we do not assume a gradient term for the concentration field in the
free energy, in equilibrium, there exists an explicit relationship between the phase field and the local concentration. Under these conditions, we need to solve the following Euler-Lagrange equation for the phase field
1 ∂  ∂φ  ∂ 2φ p' (φ )∆f [φ , c] + g ' (φ ) wT
=
,
r
+
2 ∂r  ∂r  ∂z 2
ε 2T
(4)
while in the absence of a |∇c|2 term in the free energy, the Euler-Lagrange equation for the concentration field yields a
c(φ) relationship. Accordingly, in equation (4) ∆f[φ, c(φ)] = f[φ, c(φ)] − (∂f/∂c)(c∞)[c(φ) − c∞] − f∞ is driving force of
crystallization, while properties with subscript ∞ refer to quantities characterizing the initial liquid state. Now we wish
to ensure in equilibrium (stable or unstable) that the solid-liquid interface has a fixed contact angle ψ with a foreign
wall placed at z = 0. To achieve this, we prescribe the following boundary condition at the wall, which can be viewed as
a binary generalization of Model A presented in (Gránásy et al 2007):
2∆f [φ , c(φ )]
cos(ψ ) ,
ε 2T
(n ⋅ ∇φ ) =
(5)
where n is the normal vector of the wall. The motivation for this boundary condition is straightforward in the case of a
stable triple junction, in which the equilibrium planar solid-liquid interface has a contact angle ψ with the wall. The
wall is assumed to lead to an ordering of the adjacent liquid, an effect that extends into a liquid layer of thickness d,
which is only a few molecular diameters thick (see e.g., Toxvaerd 2002, Webb et al 2003). If we take plane z = z0,
which is slightly above this layer, i.e., z0 > d, the structure of the equilibrium solid-liquid interface remains unperturbed
by the wall (see figure 1). Then along the z = z0 plane the phase field and concentration profiles are trivially related to
the equilibrium profiles across the solid-liquid interface. Evidently, in the interface the following relationship holds
ε 2T  ∂φ 
2

 = ∆f [φ , c (φ )] ,
2  ∂nSL 
(6)
where nSL is a spatial coordinate normal to the solid-liquid interface, while the component of ∇φ normal to the wall is
then (n⋅∇φ) = (∂φ/∂nSL)⋅cos(ψ) = [2∆f/(ε2T)]1/2⋅cos(ψ). (Remarkably, if in equilibrium a parabolic groove approximation
by Folch and Plapp (2003 2005) is applied for the free energy surface, one finds that conveniently ∆f[φ, c(φ)] =
wTg(φ).) While equation (5) is straightforward for the equilibrium planar solid-liquid interface, generalization of this
approach for nuclei involves further considerations. Indeed, in the undercooled state the planar interface is not in equilibrium, ∆f[φ, c(φ)] is a tilted double well, and equation (6) is not valid anymore. Note that it is the capillary pressure that
restores the uniform chemical potential inside the nucleus (being in unstable equilibrium). While, in principle, it would
be possible to solve the appropriate spherical Euler-Lagrange equation for the phase field, and use the respective solution to determine the normal component PN(φ) of the pressure tensor that makes the chemical potential spatially uniform, it seems rather unpractical. It turns out, however, that at least for large nuclei (small undercoolings) a fairly good
5
Figure 1. Typical cross-sectional phase-field map of a nucleus if the structural effects of a wall placed at z = 0 are considered (computation performed with Model B, Warren proposed in Gránásy et al (2007)). Note the boundary layers between the wall and the
solid phase (φ = 0), and between the wall and the liquid phase (φ = 1). Note also that the crystal becomes disordered at the wall, while
an ordering of the liquid takes place near the wall. Above plane z = z0 the solid liquid interface remains unperturbed by the presence
of the wall. In the case of a stable triple junction, however, the solid-liquid interface will be planar (not curved as for nuclei).
approximation can obtained if equation (5) is retained, however, with ∆f ' = ∆f − [1 − p(φ)]⋅∆f0, where ∆f0 is the driving
force of solidification in the undercooled state. Note that the correction term mimics the effect of capillary pressure.
2.2. Polycrystalline phase-field theory with quaternion representation of crystallographic orientations
Here we use the three-dimensional PF model of polycrystalline solidification (Pusztai et al 2005a 2005b). Besides the
usual square-gradient and local free energy density terms, the free energy functional consists of an orientational contribution:
 ε φ 2T

2
F = ∫ d 3r 
∇φ + f (φ , c, T ) + f ori  .
 2

(7)
The local physical state of the matter (solid or liquid) is characterized by the phase field φ and the solute concentration
c, while εφ is a constant, and T is the temperature. The local free energy density is assumed to have the form f(φ, c, T) =
w(φ)T g(φ) + [1 − p(φ)] fS(c) + p(φ) fL(c), where the intuitive “double well” and “interpolation” functions shown in the
section 2.1.1 are used, while the free energy scale is w(φ) = (1 – c) wA + c wB. The respective free energy surface has
two minima (φ = 0 and φ = 1, corresponding to the crystalline and liquid phases, respectively), whose relative depth is
the driving force for crystallization and is a function of both temperature and composition as specified by the free energy densities in the bulk solid and liquid, fS,L(c,T), taken here for the binary systems from the ideal solution model, or
from CALPHAD type computations (computer aided CALculation of PHAse Diagrams).
The orientational contribution to free energy fori has been obtained as follows. In 3D, the relative orientation with respect to the laboratory system is uniquely defined by a single rotation of angle η around a specific axis, and can be expressed in terms of the three Euler angles. However, this representation has disadvantages: It has divergences at the
poles ϑ = 0 and π, and one has to use trigonometric functions that are time consuming in numerical calculations. Therefore, we opt for the four symmetric Euler parameters, q0 = cos(η/2), q1 = c1 sin(η/2), q2 = c2 sin(η/2), and q3 = c3
sin(η/2), a representation free of such difficulties. (Here ci are the components of the unit vector c of the rotation axis.)
These four parameters q = (q0, q1, q2, q3), often referred to as a quaternion, satisfy the relationship ∑i qi2 = 1, therefore,
can be viewed as a point on the four-dimensional (4D) unit sphere (Korn and Korn 1970). (Here ∑i stands for summation with respect to i = 0, 1, 2, and 3, a notation used throughout this paper.)
The angular difference δ between two orientations represented by quaternions q1 and q2 can be expressed as cos(δ )
= ½ [Tr(R) − 1], where the matrix of rotation R is related to the individual rotation matrices R(q1) and R(q2), that rotate
the reference system into the corresponding local orientations, as R = R(q1)⋅R(q2)−1. After lengthy but straightforward
algebraic manipulations one finds that the angular difference can be expressed in terms of the differences of quaternion
coordinates: cos(δ) = 1 − 2∆2 + ∆4/2, where ∆2 = (q2 − q1)2 = ∑i ∆qi2, is the square of the Euclidian distance between the
points q1 and q2 on the 4D unit sphere. Comparing this expression with the Taylor expansion of the function cos(δ ), one
finds that 2∆ is indeed an excellent approximation of δ. Relying on this approximation, we express the orientational
difference as δ ≈ 2∆.
The free energy of small-angle grain boundaries increases approximately linearly with the misorientation of the
neighboring crystals, saturating at about twice of the free energy of the solid-liquid interface. Our goal is to reproduce
this behavior of the small angle grain boundaries. To penalize spatial changes in the crystal orientation, in particular the
presence of grain boundaries, we introduce an orientational contribution fori to the integrand in equation (1), which is
invariant to rotations of the whole system. While in 2D, the choice of the orientational free energy in the form fori =
6
HT[1 − p(φ)]∇θ (where the grain boundary energy scales with H) ensures a narrow grain boundary and describes
successfully both polycrystalline solidification and grain boundary dynamics (Kobayashi et al 1998 2000, Warren et al
2003, Gránásy et al 2002 2004a 2004b), in 3D we postulate an analogous intuitive form
{
f ori = 2 HT [1 − p(φ )] ∑i (∇qi )
}
2 1/ 2
.
(8)
It is straightforward to prove that this form boils down to the 2D model, provided that the orientational transition across
grain boundaries has a fixed rotation axis (perpendicular to the 2D plane) as assumed in the 2D formulation.
As in 2D, to model crystal nucleation in the liquid, we extend the orientation fields, q(r), into the liquid, where they
are made to fluctuate in time and space. Note that fori consists of the factor [1 − p(φ)] to avoid double counting of the
orientational contribution in the liquid, which is per definitionem incorporated into the free energy of the bulk liquid.
With appropriate choice of the model parameters, an ordered liquid layer surrounds the crystal as seen in atomistic simulations.
Time evolution of the field is assumed to follow relaxation dynamics described by the equations of motion:
  ∂I  ∂I 
δF
 −  + ζ φ ,
+ ζ φ = M φ ∇
δφ
  ∂∇φ  ∂φ 
(9a)
 ∂I 
 
 δF

 ∂I 
 v
c& = ∇M c ∇
− ζ j  = ∇ m Dc(1 − c)∇   − ∇
 − ζ j  ,
 RT
 δc

 ∂∇c 
 ∂c 
 
(9b)
φ& = − M φ
∂qi
δF
  ∂I
= −M q
+ ζ i = M q ∇
δqi
∂t
  ∂∇qi
 ∂I 
−
 ∂q  + ζ i .
i 

(9c)
Here I is the integrand of the free energy functional, vm is the molar volume, D the diffusion coefficient in the liquid,
and ζi are the appropriate noise terms representing the thermal fluctuations. (Conserved noise for the conserved fields
and non-conserved noise for the non-conserved fields (Karma and Rappel 1999).) The time scales for the fields are determined by the mobility coefficients appearing in the coarse-grained equations of motion: Mφ, Mc and Mq. These
coarse-grained mobilities can be taken from experiments and/or evaluated from atomistic simulations (see e.g., Hoyt et
al 2003). For example, the mobility Mc, is directly proportional to the classic inter-diffusion coefficient for a binary
mixture, the phase-field mobility Mφ dictates the rate of crystallization, while the orientational mobility Mq controls the
rate at which regions reorient, a parameter that can be related to the rotational diffusion coefficient and is assumed to be
common for all quaternion components. While the derivation of a more detailed final form of equations (9a) and (9b) is
straightforward, in the derivation of the equations of motion (equations 9(c)) for the four orientational fields qi(r), we
need to take into account the quaternion properties (∑i qi2 = 1) , which can be done by using the method of Lagrange
multipliers, yielding
 
∂q i

= M q ∇ HT [1 − p (φ )]

∂t
 
∇q i
[∑ (∇q ) ]
2 1/ 2
l
l


−q
q ∇ HT [1 − p (φ )]
 i ∑k k 


∇q k
[∑ (∇q ) ]
2 1/ 2
l
l

 +ζ .
i


(10)
Gaussian white noises of amplitude ζi = ζS,i + (ζL,i − ζS,i) p(φ) are then added to the orientation fields so that the quaternion properties of the qi fields are retained. (ζL,i and ζS,i are the amplitudes in the liquid and solid, respectively.)
This formulation of the model is valid for triclinic lattice without symmetries (space group P1). In the case of other
crystals, the crystal symmetries yield equivalent orientations that do not form grain boundaries. In previous works, we
(a)
(b)
(c)
Figure 2. Single crystal growth forms at various choices of the anisotropy parameters of the kinetic coefficient: (a) cube (ε1 = − 1.5,
ε2 = 0.3); (b) rhombo-dodecahedron (ε1 = 0.0, ε2 = 0.6); (c) truncated octahedron (ε1 = 0.0, ε2 = − 0.3). Here ε1 and ε2 are the coefficients of the fist and second terms in the Kubic harmonic expansion of the kinetic anisotropy.
7
have proposed that the crystal symmetries can be taken into account, when discretizing the differential operators used in
the equations of motions for the quaternion fields. Calculating the angular difference between a central cell and its
neighbors, all equivalent orientations of the neighbor have to be considered, the respective angular differences δ be calculated (using matrices of rotation R' = R⋅Sj⋅R−1, where Sj is a symmetry operator), of which the smallest δ value shall
be used in calculating the differential operator. (For cubic structure, there are 24 different Sj operators, if mirror symmetries whose interpretation in continuum models is not straightforward are omitted.)
Solving these equations numerically in three dimensions with an anisotropic interfacial free energy
3
17 
γ (n)
 3
 3
= S (n ) = 1 + ε 1  ∑i =1 ni4 −  + ε 2  ∑ i =1 ni4 + 66n12 n22 n32 −  ,
5
7 
γ0


(11)
or with an anisotropic phase-field mobility of similar form Mφ = Mφ,0S(n), one may obtain various single crystal growth
forms as exemplified in figure 2. Note that in equation (11) n = (n1, n2, n3) in the normal vector of the solid-liquid interface that can be expressed in terms of components of ∇φ.
2.2.1 Boundary conditions Unless stated otherwise, we have used periodic boundary condition in all directions. On foreign surfaces, a binary generalization of the boundary condition of Model A of (Gránásy et al 2007) has been applied
(see also section 2.1.2). We model directional solidification by imposing a temperature gradient (note the excess term
that appears because of the temperature dependent coefficient of ∇φ2) and a uniform flow velocity in the simulation
window. Foreign particles of given size and contact angle distributions of random lateral position and random crystallographic orientation were let in on the side, where high temperature liquid enters the simulation window.
2.2.2 Materials properties The polycrystalline calculations have been performed with three sets of materials parameters:
(i) For an ideal solution approximant of the Ni-Cu system, we used in previous studies (for details see Pusztai et al
2005a). (ii) For a parabolic groove approximation of the free energy (developed by Folch and Plapp 2003 2005) adopted
to the Ni-Cu system at 1574 K by Warren (2007). (iii) For the Al-Ti alloy of thermodynamic properties from a CALPHAD type assessment of the phase diagram (for details see Pusztai et al 2006).
2.2.3 Numerical solutions The equations of motion have been solved numerically using an explicit finite difference
scheme. Periodic boundary conditions were applied. The time and spatial steps were chosen to ensure stability of our
solutions. The noise has been discretized as described by Karma and Rappel (1999). A parallel codes relying on the
MPI/OpenMPI protocols have been developed.
2.3. Binary phase-field crystal model
In derivation of the binary PFC, the starting point is the free energy functional of the binary perturbative density functional theory, where the free energy is Taylor expanded relative to the liquid state (denoted by subscript L) up to 2nd
order in density difference (up to two-particle correlations) (Elder et al 2007):

ρ
F
= ∫ dr  ρ A ln AL
kT

 ρA

ρ
 − ∆ρ A + ρ B ln BL

ρ

 B


 − ∆ρ B 



,
(12)
1
− ∫∫ dr1dr2 [∆ρ A (r1 )C AA (r1 , r2 )∆ρ A (r2 ) + ∆ρ B (r1 )C BB (r1 , r2 )∆ρ B (r2 ) + 2∆ρ A (r1 )C AB (r1 , r2 )∆ρ B (r2 )]
2
where k is Boltzmann’s constant, ∆ρA = ρA − ρAL and ∆ρB = ρB − ρBL. It is assumed here that all two point correlation
functions are isotropic, i.e., Cij(r1, r2) = Cij(|r1 − r2|). Taylor expanding direct correlation functions in Fourier space up
to 4th order, one obtains Cij = [C0ij − C2ij∇2 +C4ij∇4]δ(r1 − r2) in real space, where ∇ differentiates with respect to r2 (see
Elder et al 2007). The partial direct correlation functions Cij can be related to measured or computed partial structure
factors (see e.g. Woodhead-Galloway and Gaskell 1968).
Following Elder et al (2007), we introduce the reduced partial number density differences nA = (ρA − ρAL)/ρL and nA
= (ρB − ρBL)/ρL, where ρL = ρAL + ρBL. It is also convenient to introduce the new variables n = nA + nB and (δN) = (nB −
nA) + (ρBL − ρAL)/ρL. Then, expanding the free energy around (δN) = 0 and n = 0 one obtains
[
]
n

F
t
v
w
u
L2
2
= ∫ dr  B L + BS (2 R 2 ∇ 2 + R 4 ∇ 4 ) n + n 3 + n 4 + γ (δN ) + (δN ) 2 + (δN ) 4 +
∇(δN ) + ... .
ρ L kT
3
4
2
4
2
2

(13)
Assuming substitutional diffusion between species A and B, i.e., the same M mobility applies for the two species, the
dynamics of n and (δN) fields decouple. Assuming, furthermore, that the mobility is a constant Me, the respective equations of motions have the form (Elder et al 2007):
8
∂n
δF
= M e∇ 2
∂t
δn
where
δF
δχ
=
∂I
∂χ
+ ∑ j (−1) j ∇ j
∂I
∂∇ j χ
δF
∂ (δN )
= M e∇ 2
,
∂t
δ (δN )
and
(14)
is the first functional derivative of the free energy with respect to field χ, and I is the
integrand of equation (13), while the respective effective mobility is Me = 2M/ρ2. Expanding BL, BS and R in terms of
(δN) with coefficients denoted as BjL, BjS and Rj, assuming that only coefficient B0L, B2L, B0S, R0 and R1 differ from zero,
and inserting the respective form of I into equations (14), one finds
{
{[
}
]
[
] }


B0S
L
L
2
2
3
δ
n
B
+
B
(
N
)
+
tn
+
vn
+
2 R0 + R1 (δN ) 2 ∇ 2 + R0 + R1 (δN ) 4 ∇ 4 n

0
2
∂n
2
,
= M e∇ 2 
S
∂t


B0
2
2
4
4
+
2 ∇ n R0 + R1 (δN ) + ∇ n R0 + R1 (δN )


2


{
([
])
{[
( [
]
] )}
[
]
}
 B L (δN ) n 2 + 2B0S n R0 + R1 (δN ) R1 ∇ 2 + R0 + R1 (δN ) 3 R1 ∇ 4 n
∂(δN )
= M e∇ 2  2
.
∂t
+ γ + w (δN ) + u (δN ) 3 − L2 ∇ 2 (δN )


(15a)
(15b)
These equations have been solved numerically using a semi-implicit spectral method based on operator splitting (Tegze
et al 2008) under periodic boundary condition on all sides after adding a conservative noise (a random flux) to them that
represent the thermal fluctuations with an ultraviolet cut off at the inter-atomic spacing.
2.4. Computational resources
The parallel codes developed for the phase-field end phase-field crystal models have been run on three recently built PC
clusters: two at the Research Institute for Solid State Physics and Optics, Budapest, Hungary, consisting of 160 and 192
CPU cores (80 dual core Athlon processors with 1 Gbit/s (normal Ethernet) communication, and 24 × 2 × 4 CPU core
Intel processors equipped with 10 Gbit/s fast communication (Infiniband)), respectively, and third PC cluster at the
Brunel Centre for Advanced Solidification Technology, Brunel University, West London, UK, consisting of 160 CPU
cores (20 × 2 × 4 CPU core Intel processors) and 1 Gbit/s (normal Ethernet) communication.
3. Results and discussion
3.1. Quantitative test of phase-field models of homogeneous crystal nucleation in the hard sphere system
The predicted nucleation barrier heights are presented for the usual intuitive and the Ginzburg-Landau expanded double
well and interpolation functions in figure 3 as a function of volume fraction. It has been found that the barrier heights
predicted by the PFT with physically motivated free energy (PFT/GL) gives a considerably closer agreement with direct
Figure 3. Comparison of the reduced nucleation barrier height (W*/kT) vs volume fraction relationships various phase-field models
predict for the hard sphere system without adjustable parameters. Predictions of PFT models with the intuitive (PFT/S) and with
Ginzburg-Landau expanded (PFT/GL) double-well and interpolation functions are presented. There are two curves for each PFT
model: one with the minimum (upper curve) and another with the maximum of the 10%−90% interface thickness deduced from atomistic simulations (Davidchack and Laird 1998). For comparison, direct results for W* from the Monte Carlo simulations (full
squares; Auer and Frenkel 2001a 2001b), and parameter free predictions from the droplet model of the classical nucleation theory
(CNT) are also shown.
9
results from atomistic simulations (Auer and Frenkel 2001a 2001b) than the PFT model with a free energy surface relying on the usual intuitively chosen double well and interpolation function (PFT/S). It is also remarkable that the droplet
model of the classical nucleation theory fails spectacularly. We note here that in a previous study (Gránásy et al.
2003a), we used an interface thickness determined by the envelope of the density peaks. We believe that the present
choice of δ10% − 90% ∈ [3.0σ, 3.3σ], which has been deduced from profiles for several physical properties should be more
reliable. It is worth noting also that the interfacial data from atomistic simulations might somewhat underestimate both
the interfacial free energy and the interface thickness due to the limited size of such simulations, which leads to a long
wavelength cut off in the spectrum of surface fluctuations. On the other hand, interfaces relevant to nucleation are of a
size scale that is comparable to the size scale of atomistic simulations, so one might expect here only minor error from
this source.
3.2. Structure and barrier for heterogeneous crystal nuclei in binary alloys
The structure of the heterogeneous nuclei forming at 1574 K in a NiCu liquid alloy (with a free energy surface approximated by a parabolic groove (Foch and Plapp 2003)) of composition (c − cS)/(cL − cS) = 0.2 under nominal contact angles ψ = 30°, 60°, 90°, 120°, and 170° at a horizontal wall enforced by the boundary condition equation (5) are shown
in figure 4. Note that the interface thickness is considerably smaller than the radius of curvature. Accordingly, in the
non-wetting limit (ψ → π), the height of the nucleation barrier can be approximated well with that from the classical
droplet model of homogeneous nuclei. However, towards ideal wetting the nuclei are made almost entirely of interface,
so the classical spherical cap model is expected to break down. Despite this, an analysis of the contour lines corresponding to φ = ½ gives contact angles within about 2° of the nominal (scattering with roughly this value). It is thus demonstrated that so far as the height of the nucleus is larger than the interface thickness the true contact angle falls reasonably
close to the nominal value, i.e., the boundary condition given by equation (5) can be used with confidence to simulate
surfaces of pre-defined contact angle of ψ.
It is also of interest to compare the nucleation barriers from the phase-field theory and from the classical spherical
cap model relying on a sharp interface (the homogeneous nucleus can also be obtained as doubling the barrier height for
90° contact angle). It appears that under the investigated conditions the catalytic potency factor f(ψ) = Whetero/Whomo follows closely the function f(ψ) = (1/4)[2 − 3 cos(ψ) + cos(ψ)3] from the classical spherical cap model (see rightmost
panel in figure 4). This is reasonable, since these nuclei, as mentioned above, are fairly classical since their radius of
curvature is large compared to the interface thickness.
Next, we apply this technique in phase-field simulations of heterogeneous nucleation. First, we apply it for the solidification of a single component system (only equation (9a) is solved here). Noise induced heterogeneous nucleation
has been simulated on complex surfaces of ψ = 60° including stairs, a checkerboard modulated surface, rectangular
grooves and randomly positioned spheres with random radius, while using the properties of pure Ni (figure 5). Also we
incorporate results for a non-wetting brush (ψ = 175°) protruding from a wetting surface (ψ = 60°), while at the center
of the simulated area a wetting stage (ψ = 60°) is placed that helps crystal nucleation (figure 6). A complex behavior is
Figure 4. Phase-field (upper row) and composition (lower row) maps for heterogeneous nuclei obtained by solving numerically the
respective Euler-Lagrange equation (equation (4)) as a function of contact angle ψ in the binary NiCu system at 1574 K. The size of
the calculation window is 100 × 150 nm. The contour lines in the upper row indicate phase field levels of φ = 0.1, 0.3, 0.5, 0.7, and
0.9, while the black contour line in the composition maps indicates the equilibrium composition of the solid phase cSe = 0.399112.
Here parabolic well parameters corresponding to an interface thickness of 1.76 nm and a solid-liquid interfacial free energy of 0.3623
J/m2 have been used. The classical (black) and non-classical (red) catalytic potency factors are shown on the right.
10
Figure 5. Noise induced heterogeneous crystal nucleation on complex surfaces of contact angle of 60°. From left to right: stairs,
rectangular grooves, checkerboard modulated surface, and spherical particles. (Properties of Ni have been used.)
Figure 6. Crystal nucleation and growth on a non-wetting nano-fiber brush. Note the effect of decreasing density of the brush (from
left to right) on crystallization. (For details see the text.)
seen: if the brush is dense, no nucleation is possible on the horizontal surfaces only at the central stage, and after nucleation the crystal "crawls" on the tips of the non-wetting brush. If the distance between the fibers in the non-wetting brush
increases crystal can climb down to the horizontal wetting surface, while if this distance between the non-wetting fibers
is large enough, nucleation may take place on the horizontal surface. Simulations of this kind might find application in
nano-patterning studies.
3.3. Modeling complex polycrystalline morphologies in three dimensions
3D phase-field simulations showing the nucleation and growth of crystallites of different habits (cube, rhombododecahedron, truncated octahedron, and dendritic) realized by prescribing appropriate kinetic anisotropies) illustrate
the application of quaternion field for describing crystallographic orientation in figures 7 and 8. The physical properties
of the Cu-Ni system has been used, the calculations were performed at 1574 K and at a supersaturation of S = (cL −
c)/(cL − cS) = 0.75, where cL = 0.466219, cS = 0.399112 and c are the concentrations at the liquidus, solidus, and the initial homogeneous liquid mixture, respectively. The diffusion coefficient in the liquid was assumed to be DL = 10−9 m2/s.
Dimensionless mobilities of Mφ,0 = 3.55 × 10−1 m3/Js (with an anisotropy of Mφ = Mφ,0{1 − 3ε0 + 4ε0 [(A1∇φ)x4 +
(Ay∇φ)y4 + (Az∇φ)z4]/A∇φ4}) and Mq,L = 8.17 m3/Js, and Mq,S = 0 were applied, while DS = 0 was taken in the solid.
The kinetics of multi-grain dendritic solidification has been simulated in a cube of 8.4 µm × 8.4 µm × 8.4 µm for a time
interval of ~ 0.16 ms. The evolution of the normalized crystalline fraction X has been analyzed in terms of the Johnsonp
Mehl-Avrami-Kolmogorov kinetics (Christian 1981), X = 1 − exp − (t / τ ) AK , where τ is a constant related to the nucleation and growth rates, and pAK is the Avrami-Kolmogorov exponent characteristic to the mechanism of transformation. The kinetic exponent evaluated from our simulations, pAK = 2.922 ± 0.001 (τ = 5 × 10−5 s), falls between those for
{
}
Figure 7. Polycrystalline structures formed by nucleation and growth of cubic, rhombo-dodecahedral and truncated octahedral crystals (from left to right, respectively). The computations have been performed on a 400 × 400 × 400 grid for ideal solution NiCu thermodynamics at 1574 K and supersaturation S = 0.8, however, with kinetic anisotropies given in the caption of figure 2.
11
Figure 8. Polycrystalline structure formed by nucleation and dendritic growth in a NiCu alloy (whose thermodynamic properties
were approximated by the ideal solution model) at 1574 K and S = 0.78, while assuming cubic crystal symmetries. Equations (9)
have been solved numerically on a 640 × 640 × 640 grid (~262 million grid points) by solving numerically the equations of motion (6
stochastic partial differential equations). The computation took about a month on 80 processors. By the end of the simulation, about
180 crystalline particles formed. Different colors indicate different crystallographic orientations.
nucleation with diffusion controlled (pAK = 2.5) and with steady state growth (pAK = 4) (see Christian 1981). This implies that some of the particles have not yet reached the fully grown steady state dendritic morphology, as is apparent in
figure 8. Larger simulations are planned to clarify further the relationship between morphology and pAK. Note that here
we have a reasonable statistics for nucleation, as by the end of the simulation, about 180 dendritic particles formed, a
number considered sufficient for such purposes (Pusztai and Gránásy 1998).
As discussed in detail in previous work (Gránásy et al 2003c 2004a 2004b 2005), reduction of the orientation mobility in the case of needle crystals may lead to the formation of Category 2 spherulites, that start to grow as a single
needle crystal but later the ends splay out and form eventually a space filling roughly spherical polycrystalline structure.
A similar transition can be seen when increasing the driving force of solidification. As demonstrated in figure 9, the
Figure 9. From needle crystal to spherulites in a phase-field theory relying on a quaternion representation of the crystallographic
orientation. The simulations have been performed by solving equations (9) on a 200 × 200 × 400 grid assuming ideal solution thermodynamics. A large kinetic anisotropy favoring a needle crystal form, characterized by the parameter values ε0 = 1/3 and A = (0, 0,
1) has been applied. The driving force of solidification increases from the left to right (S = 1.8, 1.9, 2.0, and 2.1, respectively).
12
Figure 10. Phase-field simulation of polycrystalline solidification of the Al0.45Ti0.55 alloy in a moving frame (V = 1.26 cm/s) and a
constant temperature gradient (∇T = 1.12 × 107 K/m). Composition (on the left) and orientation maps (on the right) corresponding to
times t = 2.3, 2.6 and 2.9 ms are shown. Note that the orientations corresponding to 0 and 1 are equivalent. The computation has been
performed by solving the 3D model (equations (9)) in 2D on a 600 × 3000 grid (3.93 µm × 19.69 µm). White spots in the chemical
maps indicate the foreign particles, whose diameter varies in the 13 nm − 66 nm range, and have a contact angle of ψ = 60°.
frequency by which new grains form at the ends of the needle crystal increases strongly with increasing supersaturation.
The mechanism, by which the new grains form, is via quenching orientational defects into the solid, which defects
might be identified as bunches of dislocations, as in 2D simulations (Gránásy et al 2006).
3.3.1 Modeling of directional solidification In order to model columnar to equiaxed transition (CET) in the framework
of the EU FP6 IMPRESS project (Jarvis and Voss 2005), we have extended our 3D model to describe polycrystalline
solidification of the Al0.44Ti0.55 alloy in temperature gradient and a moving frame. To enable large scale simulations, we
have used a broad interface (65.6 nm), however, included an anti-trapping current (Kim et al 1999, Karma 2001, Kim
2007) to ensure a quantitative description of dendrites. In the simulation window, the material is made to move with a
homogeneous velocity from the bottom to the top, while a fixed temperature gradient is prescribed in the vertical direction. Particles of given number density, random orientation and size, and of given contact angle are let to enter into the
simulation window at the bottom edge. Snapshots of the chemical and orientation maps illustrating polycrystalline solidification under such conditions are presented in figure 10. As a result of the interplay between heterogeneous nucleation and growth, after the initial transient, we observe stochastically alternating nucleation-controlled and growthcontrolled periods. This is a non-steady solution appearing in the CET zone. A detailed analysis of this phenomenon
will be presented elsewhere (Pusztai et al 2008).
3.3.2 Phase separation and polycrystalline solidification in the presence of fluid flow In order to address the solidification of Al-Bi monotectic alloys (candidates for a new generation of self lubricating bearing materials), the 2D version of
our polycrystalline phase-field theory has been combined with viscous flow. Solidification has been then modeled via
introducing a phase-field dependent viscosity, and a non-classical stress-tensor related to the phase-, composition-, and
orientation fields (Tegze and Gránásy 2006), while the regular solution model has been used to approximate the thermodynamics of the Al-Bi system. Inside the liquid-liquid immiscibility region we observed various hydrodynamic effects (Tegze et al 2005). Besides the solutal- and thermocapillary motion, we have seen flow assisted coagulation and
bicontinuous phase separation (figure 11), mechanisms identified by Tanaka and coworkers (Tanaka 1995 1996, Tanaka
13
(a)
(b)
(c)
(d)
Figure 11. Liquid phase separation and solidification in monotectic alloys (regular solution approximant of Al-Bi): (a) Collision
assisted collision of liquid droplets (cBi = 0.25, T = 920 K, 250 × 250 section of a 512 × 512 grid); (b) bicontinuous phase separation
(cBi = 0.5, T = 900 K, 512 × 512 grid); (c),(d) solidification of phase separating liquid (cBi = 0.23, T = 750 K, 512 × 512 section of a
1024 × 1024 grid). Composition (a) − (c) and orientation maps (d) are shown. In panels (a) − (c) arrows indicate the velocity field.
and Araki 1998). It has also been found that the solute pile up ahead of the solidification front might significantly accelerate droplet nucleation in the metastable region of the liquid-liquid coexistence region (figure 11).
3.4. Atomistic simulations for polycrystalline solidification of a binary alloy in two dimensions
We have performed simulations for the PFC model on a 16,384 × 16,384 grid using the same model parameters as Elder
et al (2007), however, with half of their spatial step. Accordingly, our simulation window contains roughly 1.6 million
atoms. Solidification has been initiated by inserting 5, 50 and 500 randomly oriented and positioned crystalline clusters
Figure 12. Polycrystalline solidification in the binary phase-field crystal model (the distribution of the (δN) field is shown). 1st row:
Dendritic growth of 5 crystalline particles (snapshots taken at 1,000, 5,000, 10,000 and 20,000 time steps are shown). 2nd row:
growth of 50 particles (snapshots taken at 1,000, 3,000, 5,000 and 10,000 time steps are shown). 3rd row: growth of 500 particles
(snapshots taken at 250, 500, 750 and 1,500 time steps are shown). The simulations have been performed on a 16,384 × 16,384 grid,
using a semi-implicit spectral method. Note that here the position of all atoms of the crystalline phase are known accurately.
14
Figure 13. Crystallization kinetics for binary phase-field crystal simulations shown in figure 12. (a) Number of atoms in the crystalline phase vs number of time steps; (b) Avrami plots (X and Xmax are the transformed fraction and its maximum; the slope of the
curve is the Avrami-Kolmogorov exponent pAK); and (c) the kinetic (Avrami-Kolmogorov) exponent as a function of the reduced
transformed fraction. (Upward and downward pointing triangles and squares correspond to 500, 50 and 5 particles, respectively.)
of 13 atoms each into the simulation window. The resulting multi-grain structures are shown in figure 12 (snapshots of
the "composition field" (δN) are displayed). The respective time dependencies of the number of atoms in the crystalline
phase are presented in figure 13(a). The latter quantity has been obtained by counting the atoms in the crystalline state
(an atom has been supposed to belong to the solid phase if its density peak was larger than the average of the value for
the bulk liquid and the maximum value for the bulk crystal) using the public domain software ImageJ (Abramoff et al
2004). The higher level of crystalline fraction observed in the 500 particle simulation (~1.5 million atoms of a total of
~1.6 million) signals a more efficient solute trapping, probably attributable to fact that here the initial transient of fast
growth rate represents a larger fraction of the total solidification time than for 50 or 5 particles. This is also consistent
with the observation that the contrast of the (δN) field grows with time. The time evolution of crystallization has been
analyzed in terms of the JMAK kinetics. The respective Avrami plots and the kinetic exponent vs reduced transformed
fraction curves are displayed in figures 13(b) and (c). The Avrami plots are not linear, and the respective AvramiKolmogorov exponents (pAK) vary with the transformed fraction (or time). Apart from an initial transient, the observed
pAK values fall between the limiting values pAK = d/2 = 1 and pAK = d = 2 corresponding to diffusion controlled (conserved dynamics) and interface controlled growth of fixed number of particles in 2D (Christian 1981). Possible origin of
the observed time dependencies of pAK is that due to mass conservation, and the differences in the densities of the crystal and liquid, the driving force for crystallization decreases as crystallization proceeds. Screening effects characteristic
to highly anisotropic growth (Shepilov 1990, Shepilov and Baik 1994, Birnie and Weinberg 1995, Pusztai and Gránásy
1998) are also expected to influence transformation kinetics of the dendritic particles. Finally, we note that the behavior
of the pAK(X) curve for the 5 dendritic particles reflects the small number of these particles, which cannot provide a satisfactory statistics for an accurate evaluation of the kinetic exponent. Unfortunately, significantly larger simulations for
a large number of fully developed dendrites cannot be easily made with the present numerical technique and the hardware we used.
4. Summary
Using various phase-field techniques, we have addressed diverse aspects of polycrystalline solidification, including homogeneous and heterogeneous nucleation of growth centers, and polycrystalline growth. Along these lines, we have
shown that using a physically motivated (Ginzburg-Landau expanded) free energy in the phase-field approach, a reasonably accurate prediction can be obtained for the nucleation barrier of homogeneous crystal nucleation in the hard
sphere system. We have then presented a method for incorporating walls of pre-defined contact angle into phase-field
simulations, and demonstrated that rather complicated problems (heterogeneous nucleation on patterned surfaces/nanofiber brush) can be treated this way. Next, we have shown that phase-field models based on a quaternion representation
of the crystallographic orientation are able to address the formation of fairly complex three dimensional polycrystalline
structures, including multi-grain dendritic solidification and the formation of polycrystalline spherulites. The effect of
temperature and flow fields on polycrystalline solidification has also been explored. Finally, we have used a recently
developed atomistic approach, the "phase-field crystal" model, to investigate multi-grain dendritic crystallization in a
binary liquid alloy. We believe that these modeling tools and their descendants/combinations supported by atomistic
simulations and ab initio computations will find application in various branches of materials science and technology.
15
Acknowledgments
L. G. thanks J. A. Warren and J. F. Douglas for discussions on heterogeneous nucleation and polycrystalline solidification, M. Plapp for illuminating discussions on noise, and K. R. Elder for communications regarding the PFC method.
This work has been supported by contracts OTKA-K-62588, ESA PECS Nos. 98021, 98043 and 98059, and by the EU
FP6 Project IMPRESS under Contract No. NMP3-CT-2004-500635. P. T. is a grantee of the Bolyai János Scholarship
of the Hungarian Academy of Sciences. G. B. has been supported by an EPSRC grant.
References
Abramoff M D, Magelhaes P J and Ram S J 2004 Biophotomics Int. 11 36
Athreya B P, Goldenfeld N and Dantzig J A 2006 J. Stat. Phys. 125 1019
Athreya B P, Goldenfeld N, Dantzig J A, Greenwood M and Provatas N 2007 Phys. Rev. E 76 art no 056706
Auer S and Frenkel D 2001a Nature 409 1020
Auer S and Frenkel D 2001b Nature 413 71
Beltram-Sanchez L and Stefanescu D M 2004 Metall. Mater. Trans. A 35, 2471
Birnie III D P and Weinberg M C 1995 J. Chem. Phys. 103 3742
Boettinger W J, Warren J A, Beckermann C and Karma A 2002 Ann. Rev. Mater. Res. 32 163
Börzsönyi T, Tóth-Katona T, Buka Á and Gránásy L 1999 Phys. Rev. Lett. 83 1853

2000 Phys. Rev. E 62 7817
Bouissou Ph, Chiffaudel A, Perrin B and Tabeling P 1990 Europhys. Lett. 13 89
Broughton J Q and Gilmer G H 1986 J. Chem. Phys. 84 5749
Cacciuto A, Auer S and Frenkel D 2003 J. Chem. Phys. 119 7467
Cahn J W and Hilliard J E 1958 J. Chem. Phys. 28 258
Cahn R V 2001 The Coming of Materials Science (Oxford: Pergamon)
Chen L Q 2002 Ann. Rev. Mater. Res. 32 113
Cheng Z, Chaikin P M, Zhu J, Russel W B and Meyer W V 2002 Phys. Rev. Lett. 88 art no 015501
Christian J W 1981 Transformations in Metals and Alloys (Oxford: Pergamon)
Davidchack R L and Laird B B 2000 Phys. Rev. Lett. 85 4751

1998 J. Chem. Phys. 108 9452
Davidchack R L, Morris J R and Laird B B 2006 J. Chem. Phys. 125 art no 094710
Diepers H J, Ma D and Steinbach I 2002 J. Cryst. Growth 237-239 149
Elder K R, Katakowski M, Haataja M and Grant M 2002 Phys. Rev. Lett. 88 art no 245701
Elder K R and Grant M 2004 Phys. Rev. E 70, art no 051605
Elder K R, Provatas N, Berry J, Stefanovic P and Grant M 2007 Phys. Rev. B 75, art no 064107
Fan D and Chen L Q 1996 Acta Mater. 45, 611
Folch R and Plapp M 2003 Phys. Rev. E 68 art no 010602
Folch R and Plapp M 2005 Phys. Rev. E 72 art no 011602
Geil P H 1963 Polymer single crystals (New York: Wiley)
Goldenfeld N, Athreya B P and Dantzig J A 2005 Phys. Rev. E 72, art no 020601
Gránásy L and Pusztai T 2002 J. Chem. Phys. 117 10121
Gránásy L, Börzsönyi T and Pusztai T 2002 Phys. Rev. Lett. 88 art no 206105
Gránásy L, Pusztai T, Tóth G, Jurek Z, Conti M and Kvamme B 2003a J. Chem. Phys. 119 10376
Gránásy L, Pusztai T, Warren J A, Börzsönyi T, Douglas J F and Ferreiro V 2003b Nature Mater. 2 92
Gránásy L, Börzsönyi T and Pusztai T 2003c Interface and Transport Dynamics, Computational Modelling, ed H Emmerich, B Nestler and M Schreckenberg, Lecture Notes in Computational Science and Engineering, Vol. 32, (Berlin: Springer) p 190
Gránásy L, Pusztai T and Warren J A 2004a J. Phys.: Condens. Matter 16 R1205
Gránásy L, Pusztai T, Börzsönyi T, Warren J A and Douglas J F 2004b Nature Mater. 3 635
Gránásy L, Pusztai T, Tegze G, Warren J A and Douglas J F 2005 Phys. Rev. E 72 art no 011605
Gránásy L. Pusztai T, Tóth G I, Tegze G, Warren J A and Douglas J F 2006 Philos. Mag. 86 3757
Gránásy L, Pusztai T, Saylor D and Warren J A 2007 Phys. Rev. Lett. 98 art no 035703
Gunton J D, San Miguel M and Sahni P 1983 Phase Transitions and Critical Phenomena ed C Domb and J L Lebowitz (London:
Academic) vol. 8, p 267
Howe J M, 1996 Philos. Mag. A 74 761
Howe J M and Saka H 2004 MRS Bull. 29 951
Hoyt J J, Asta M and Karma A 2003 Mater. Sci. Eng. Rep. R 41 121
Huisman W J, Peters J F, Zwanenburg M J, de Wries S A, Derry T E, Abernathy D and van der Veen J F 1997 Nature 390, 379
Hyde S T, Carnerup A M, Larsson A K, Christy A G and Garcia-Ruiz J M 2004 Physica A 339 24
Jacot A and Rappaz M 2002 Acta Mater. 50 1909
Jarvis D J and Voss D 2005 Mater. Sci. Eng 413-414 583
Jin L-W, Claborn K A, Kurimoto M, Geday M A, Maezawa I, Sohraby F, Estrada M, Kaminsky W and Kahr B 2003 Proc. Natl.
Acad. Sci. 100 15297
Karma A 2001 Phys. Rev. Lett. 87 art no 115701
Karma A and W.-J. Rappel W-J 1999 Phys. Rev. E 60, 3614
16
Khan S R, Finlayson B and Hackett R L 1979 Investigative Urology 17 199
Kim S G 2007 Acta Mater. 55 4391
Kim S G, Kim W T and Suzuki T 1999 Phys. Rev. E 60 7186
Kobayashi R and Warren J A 2005a TMS Lett. 2 1

2005b Physica A 356 127
Kobayashi R, Warren J A and Carter W C 1998 Physica D 119 415

2000 Physica D 140 141
Korn G A and Korn T M 1970 Mathematical Handbook for Scientists and Engineers (New York: McGraw-Hill)
Koss M B, LaCombe J C, Chait A, Pines P, Zlatkowski M, Glicksman M E and Kar P 2005 J. Cryst. Growth 279 170
Krebs M H R, MacPhee C E, Miller A F, Dunlop I E, C. M. Dobson C M and A. M. Donald A M 2005 Biophys. J. 88 2013
Krill C E and Chen L Q 2002 Acta. Mater. 50 3057
Laird B B and Haymet A D J 1992 Chem. Rev. 92 1819
Lambert C C, Lambert G, Crundwell G and Kantardjieff K 1998 J. Exp. Zoology 282 323
Lewis D, Pusztai T, Gránásy L, Warren J and Boettinger W 2004 J. Metals. 56 34
Li L, Li C Y and Ni C 2006 J. Am. Chem. Soc. 128 1692
Li S W, Li X R, Lowengrub J and Glicksman M E 2007 Fluid Dyn. Mater. Processing 2 1
Magill J H 2001 J. Mater. Sci. 36 3143
Morin B, Elder K R, Sutton M and Grant M 1995 Phys. Rev. Lett. 75 2156
Murray B T, Wheeler A A and Glicksman M E 1995 J. Cryst. Growth 154 386
Napolitano R E, Meco H and Jung C 2004 J. Metals 56 16
Oxtoby D W 1991 Liquids, Freezing and Glass Transition ed J P Hansen et al. (Amsterdam: Elsevier) p 145
Provatas N, Dantzig J A, Athreya B, Chan P, Stefanovic P, Goldenfeld N and Elder K R 2007 J. Metals 59 83
Pusztai T and Gránásy 1998 Phys. Rev. B 57 14 110
Pusztai T, Bortel G and Gránásy L 2005a Europhys. Lett. 71 131

2005b Mater. Sci. Eng. A 413-414 412

2006 Modeling of Casting, Welding and Advanced Solidification Processes - XI ed C A Gandin and M Bellet (Warrendale:
The Minerals, Metals and Materials Society) p 409
Pusztai T, Környei L, Bansel G and Gránásy L 2008 unpublished
Qian X and Cummins H Z 1990 phys. Rev. Lett.64 3038
Ramalingam H, Asta M, van der Walle A and Hoyt J J 2002 Interf. Sci. 10 149
Schmidt A 1996 J. Comp. Phys. 125 293
Shelf C A and Hill C A 2003 J. Cave and Karst Studies 65 130
Shepilov M P 1990 Sov. Phys. Crystallogr. 35 164
Shepilov M P and Baik D S 1994 J. Non-Cryst. Solids 171 141
Shih W H, Wang Z Q, Zeng X C and Stroud D 1987 Phys. Rev. A 35 2611
Steinbach I, Pezzola F, Nestler B, Seesselberg M, Prieler M, Schmitz G J and Rezende J L L 1996 Physica D 94 135
Steinbach I, Beckermann C, Kauerauf B, Li Q and Guo J 1999 Acta Mater. 47 971
Tan J and Zabaras N 2006 J. Comp. Phys. 211 36

2007 J. Comp. Phys. 226 131
Tanaka H 1995 J. Chem. Phys. 103 2361

1996 J. Chem. Phys. 105 10909
Tanaka H and Araki T 1998 Phys. Rev. Lett. 81 389
Tegze G and Gránásy L 2006 Proc. Modeling of Cating, Welding and Advanced Solidification Processes - XI ed C A Gandin and M
Bellet (Warrendale: The Minerals, Metals and Materials Society) p 513
Tegze G, Pusztai T and Gránásy L 2005 Mater. Sci. Eng. A 413-414 418
Tegze G, Bansel G, Gránásy L, Tóth G I and Pusztai T 2008 unpublished
Tiaden J, Nestler B, Diepers H J and Steinbach I, 1998 Physica D 115, 73
Toxvaerd S 2002 J. Chem. Phys. 117 10303
Tupper P F and Grant M 2008 Europhys. Lett. 81 art no 40007
Tryggvason G, Brunner B, Esmaeeli A, Juric D, Al-Rawahi N, Tauber W, Han J, Nas S and Jan Y-J 2001 J. Comput. Phys. 169 708
van der Veen J F and Reichert H 2004 MRS Bull. 29 958
Wang S L, Sekerka R F, Wheeler A A, Murray B T, Coriell S R, Braun R J and McFadden G B, 1993 Physica D 69, 189
Warren J A 2007 personal communication
Warren J A, Kobayashi R, Lobkovsky A E and Carter W C 2003 Acta Mater. 51 6035
Webb E B, Grest G S and Heine D R 2003 Phys. Rev. Lett. 91 art no 236102
Woodhead-Galloway J and Gaskell T 1968 J. Phys. C 1 1472
Zacharaissen K E and Hammel H T 1988 Cryobiology 25 143
Zhu M-F, Dai T, Lee S-Y and Hong C-P 2008 Comp. Math. Appl. 55 1620
Zhu M-F and Hong C-P 2002 Phys. Rev. B 66 art no 155428
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